Duality and domains in supersymmetric gauge theories

Abstract

This thesis investigates duality properties of four-dimensional N = 2 supersymmetric Yang-Mills theory. By restricting the domain of the effective coupling to an appropriate fundamental domain the order parameter on the Coulomb branch becomes one-to-one. For special cases, such as the pure SU(2) theory, this domain corresponds to the fundamental domain of a certain subgroup of SL(2,Z). In general, the story is more subtle. For example, including matter introduces branch points in the domains and they typically do not correspond to those of any subgroup of SL(2,Z). Despite this, we develop new and general tools to analyse the Coulomb branch and construct fundamental domains for the effective coupling. In certain limits of the masses, singularities can merge or decouple, and the domains undergo transitions. The branch points then provide a natural mechanism for this evolution. For the pure theory with gauge group SU(3) we show that on certain subloci of the Coulomb branch the order parameters can again be written as combinations of elliptic modular forms. The fundamental domains for the running couplings play an important role as integration domains, when calculating correlation functions of topologically twisted N = 2 theories. We study these connections for the SU(2) theories with 0 ≤ Nf ≤ 3 fundamental hypermultiplets.